Contents
Advanced granulation theory at particle level
Contents
Advanced granulation theory at particle level
1.
1.1
1.2
1.3
1.3.1
1.3.1.1
1.3.1.2
1.3.1.3
1.3.1.4
1.3.1.5
1.3.2
1.3.2.1
1.3.2.2
1.3.3
3
Fluid bed agglomeration at particle level
Mechanisms involved in the growth rate of granules
Wetting and nucleation
Granule growth behaviour and kinetics
Mechanical properties of liquid-bound granules
Interparticle forces
Static strength – capillary and surface tension forces
Dynamic strength - viscous forces
Breakage and attrition of wet granules
Wet granule strength summary
Granule growth modelling – Class I and Class II models
Class I models: Coalescence of non-deformable granules
Class I models: Coalescence of deformable granules
Granule growth regimes
4
4
5
8
8
11
12
18
20
22
23
26
29
34
Summary
38
Table of symbols
39
Literature
45
Endnotes
61
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Advanced granulation theory at particle level
Advanced granulation theory at particle level
The present text concerns the micro-level (particle-level) perspective on the different stages of the
granulation process. A range of the newest and advanced quantitative models is presented hereby
introducing recent advances in wetting and nucleation modelling, and theory describing granule
growth behaviour. The different bonding mechanisms and the strength of liquid bonded particles
are emphasised and recent advances in simulation of wet granule breakage is reviewed. Further,
some of the more advanced coalescence models are introduced with primary focus on class I
models accounting for coalescence of non-deformable as well as deformable granules.
The text is aimed at undergraduate university or engineering-school students working in the field
chemical and biochemical engineering as well as particle technology. Newly graduated as well as
experienced engineers may also find relevant new information as emphasis is put on the newest
scientific discoveries and proposals presented in the last few years of scientific publications. It is
the hope that the present text will provide a complete and up-to-date image of how far modern
granulation theory has come, and also further provide the reader with qualitative rules of thumb
that may be essential when working with granulation processes. The comprehensive literature list
may also hopefully be an inspiration for further reading.
I alone am responsible for any misprints or errors and I will be grateful to receive any critics
and/or suggestions for further improvements.
Copenhagen, September 2006
Peter Dybdahl Hede
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
1. Fluid bed agglomeration at particle level
An understanding of the various processes and mechanisms in the granulation process at microor particle-level is essential in any type of modelling approach. If the particle-level mechanisms
are not fully understood, an adequate modelling of the entire system at meso- or macro-scale does
not have a fair chance of success. The following text focuses is on the particle-level modelling of
some of the most important processes that may take place during wet granulation.
1.1 Mechanisms involved in the growth rate of granules
Fluidised bed granulation is sometimes referred to as a one-pot system as the elementary steps of
the process occur in the same chamber. Fluidisation and mixing of the solid bulk are provided by
an upward hot air flow. Fine droplets of liquid solvent with binder material are distributed by the
nozzle. As the droplets come into contact with solid particles, a liquid layer forms at the particle
surface. When a wet particle collides with another particle in the fluid bed a liquid bridge appears
between the two particles. When subsequent drying occurs, the solvent evaporates and a solid
bridge arises due to the solidification of the binder material. The repetition of these steps causes
growth of the fluidised bed particles through agglomeration until a point where growth is
counteracted by breakage due to insufficient liquid binder material (Turchiuli et al., 2005 and
Iveson et al., 2001a). Formally, these different steps can be divided into three principal
mechanisms being: wetting and nucleation, agglomeration and growth by layering, and finally,
breakage and attrition (Iveson et al., 2001a and Cameron et al., 2005).
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
In respect to the modelling of the agglomeration and coating process, it is obviously the
mechanisms associated with agglomeration that have the primary interest. Hence, the primary
focus in the following chapter concerns advanced modelling aspects of wet granule
agglomeration and theory describing the mechanical properties of wetted particles. The
phenomena associated with wetting and nucleation were extensively covered in Hede (2005 &
2006b) and only some of the latest approaches will be presented in the present document. For
more fundamental information on nucleation, Hapgood (2000) and Wauters (2001) should be
consulted. Likewise will breakage and attrition of dry granules not be covered as these topics
were covered extensively in Hede (2005 & 2006b) besides being reviewed lately by Reynolds et
al., (2005).
1.2 Wetting and nucleation
The initial step in the wet agglomeration processes is the process of bringing liquid binder into
contact with the particles powder and attempt to distribute this liquid evenly throughout the
fluidised particles. This is usually referred to as wetting. In batch granulation, nucleation refers to
the formation of initial aggregates in the beginning of the granulation process and the formed
nuclei provide the initial granular stage for further agglomeration (Cameron et al., 2005). Only in
the last few years, the effects of nucleation on the final product properties have been recognised
and within the last two years more advanced approaches have been introduced. Litster et al. (2001)
presented the dimensionless spray flux as a measure of the density of droplets landing on a
particle bed surface. The dimensionless spray flux1 is used as a tool to predict the controlling
mechanism of the nucleation process. Hapgood et al. (2004) extended this work using a Monte
Carlo2 model to predict the extent of droplet overlap in the spray zone and therefore the
proportion of droplets that produces single nuclei. Work by Hapgood et al. (2003) and Litster
(2003) further introduced the nucleation regime map3 being capable of predicting the controlling
nucleation mechanism as a function of the dimensionless spray flux and the liquid droplet
penetration time Wd divided by the particle circulation time4 Wc. This nucleation regime map is to
some extent capable of describing previously reported data by Tardos et al. (1997) but the
original dimensionless spray is not adequate enough to predict and describe any full nuclei size
distribution, which nevertheless is a prerequisite if the nucleation regime map should have any
practical importance. This is due to the fact that the original dimensionless spray flux does not
take into account that a single nucleus formed from a single droplet is larger than the original
droplet due to the extra volume of the solids. Therefore the fraction particle bed coverage of
nuclei will be higher than the fraction particle bed coverage of the droplets from which they are
made (Wildeboer et al., 2005).
Schaafsma et al. (1998 & 2000a) defined in accordance with Hapgood et al. (2004) the nucleation
ratio5 J as the ratio of the volume (or mass) of a nucleus granule formed to the volume (or mass)
of the droplet. However, in the case such a nucleation ratio should have any relevance for
practical nucleation or agglomeration purposes it is the projected area of the granules that have
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
the primary interest. In recent work by Wildeboer et al. (2005) they introduced another similar
parameter being the nucleation area ratio according to:
Ka
an
ad
(1.1)
where an is the projected area of the nuclei granules and ad is the projected area of the binder
droplets. Although not very likely the case, Wildeboer et al. (2005) assumed for simplicity that
the nucleation area ratio is droplet-size independent and thereby constant. The probability of a
single droplet forming a single nucleus will then relate to Ka\a rather than to \a as in the original
approach by Litster (2003) and Hapgood et al. (2003). As an extension of the original
dimensionless spray flux expression, Wildeboer et al. (2005) suggested a dimensionless spray
number according to:
n
Ka ˜ a
˜K
3˜ V
a
2 ˜ A ˜ dd
(1.2)
is the powder flux through the spray zone, V
the volumetric spray rate of spherical
where A
droplets produced by the nozzle and dd is the liquid droplet diameter.
What is also of importance regarding the nucleation formation is the distribution of the liquid
binder mass underneath the spray zone. Experiments in rotating drum granulator by Wauters et al.
(2002) indicated that the density of liquid binder mass is highest in the center underneath the
spray and decreases further away from the center. This means that the assumption of uniform
droplet distribution across the width of the spray zone is problematic. In a new approach by
Wildeboer et al. (2005) the spray zone is represented by a one-dimensional flat fan spray where
the binder liquid distribution along the direction of particle movement (x direction) is projected
onto the center line of the spray6 instead of being assumed uniformly distributed. With this
approach, any type of nozzle with its own typical two-dimensional binder liquid distribution can
be represented. Based on the data by Wauters et al. (2002), a normal distribution was fitted for
which it was seen that such a distribution describes the liquid binder distribution well. One
problem in representing the spray distribution with a normal distribution is that there will be loss
of binder mass outside the finite width of the spray zone. To account for this, Wildeboer et al.
(2005) defined a dimensionless nuclei distribution function along the width of the spray zone (y
direction) given by a truncated normal distribution according to:
n (y)
N(y, mean , width )
­
˜ W ˜ Ka ˜ °
® P( 0.5 ˜ W y 0.5 ˜ W)
°0
¯
for - 0.5 ˜ W y 0.5 ˜ W
(1.3)
elsewhere
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
in which to \n(y) is the local dimensionless nucleation function and relates directly to the local
probability of nuclei overlap at position y in the spray zone. P(-0.5W < y < 0.5W) is the
probability of a droplet from distribution N(y, mean, width) falling within the defined spray zone
of width W. Wildeboer et al. (2005) chose W so that P > 0.95. N(y, mean, width) is a simple
Gaussian distribution according to:
N(y, mean , width )
1
width 2
exp(½ ˜ ((y mean ) 2 / width ) 2 )
(1.4)
where mean is the mean in the Gaussian distribution and width is the standard deviation of liquid
binder spread along the width of the spray zone.
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Based on the developed model in equation 1.3 Wildeboer et al. (2005) performed a number of
Monte Carlo simulations thereby simulating a real spray of liquid binder droplets and the
formation of nuclei accounting for droplet overlap. It was observed that the effects of the liquid
binder flow rate and the velocity of particles perpendicular to the width of the spray zone are the
same, as both parameters affect only the density of droplets on the particle bed without changing
the individual droplet properties. Changes in droplet diameter dd obviously changed the number
and volume of the droplets but as the variation in dd does not change the total volume of the
nuclei produced, the effect on particle size distribution was observed to be quite small. The
parameter Ka on the other hand does change the total volume of the nuclei produced and hence Ka
was observed to have a large effect on the particle size distribution. Simulations clearly indicate
its importance regarding the control of the produced particle size distribution.
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Although the wetting and nucleation step may be seen as a minor part of the granulation process it
is nevertheless a vital part of the process, and spray rate conditions and particle flux in the spray
zone has primary importance for the entire process and the resulting granule properties. The
current work by Wildeboer et al. (2005) and Wauters et al. (2002) makes it possible to simulate
the nuclei size distribution based on relevant process parameters with adequate precision. The
model by Wildeboer et al. (2005) may be used to model the spray zone where partially wetted
particles are presented to the spray. This further makes the model somewhat suitable for replacing
the traditional nucleation term in one-dimensional population balance models which will be
introduced in chapter three. Implementation of fundamental knowledge of nuclei formation and
wetting conditions may lead to predictions of nuclei size, porosity- and moisture distributions
which are all vital properties in respect to the quality of the final granules.
1.3 Granule growth behaviour and kinetics
Granule growth occurs whenever the wetted particles in the fluid bed collides and sticks
permanently together. For two large granules this process is traditionally referred to as
coalescence or simply agglomeration. The sticking of fine material onto the surface of large preexisting granules is sometimes referred to in old articles as layering but (e.g. Kapur & Fuerstenau,
1969) but as the distinction between layering and coalescence depends on the chosen cut-off size
used to demarcate fines from granulates, agglomeration or coalescence are often the only terms
used. Nowadays layering is used as a synonym for coating being growth due to droplet impact
only (Iveson et al., 2001a).
Whether or not a collision between two granules results in permanent coalescence depends on a
wide range of factors including the mechanical properties of the granules and the availability of
liquid binder at or near the surfaces of the granules. Being a complex phenomenon,
agglomeration has traditionally been treated qualitatively and quite a lot of articles exist in which
the influence of different factors on agglomeration tendency has been treated qualitatively as it
has been reviewed by Hede (2006b).
In the process towards a full quantitative description of the agglomeration process the
agglomeration situation must necessarily be somewhat simplified. The majority of models
treating agglomeration at particle level analyses the situation by viewing the granulation situation
between two particles. This allows detailed studies of mechanical properties as well as collision
studies far from the chaotic situation inside fluid beds. This naturally limits the applicability
regarding the description of the entire agglomerating system in real fluid beds, but as it will be
emphasised in later chapters much vital information for the use in macro-scale models can in fact
be achieved from simplified particle-level studies.
1.3.1 Mechanical properties of liquid-bound granules
An agglomerate can exist in a number of different spatial structures depending on the binder
liquid saturation. It is the amount of liquid binder as well as the humidity conditions in the bed
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
that determines the degree of saturation, which again determines the spatial structure of the final
granule (Jain, 2002). Such wet liquid bridges are obviously only temporary structures and more
permanent bonding within the granule is created by solid bridges formed as solvent evaporates
from the bridges during further fluidisation. Solid bridges between particles may take basically
three forms: crystalline bridges, liquid binder bridges and solid binder bridges. If the material of
the particles is soluble in the binder liquid, crystalline bridges may be formed when the liquid
evaporates. The process of evaporation reduces the proportion of liquid in the granules again
producing high strength pendular bridges before crystals form. Alternatively, the dissolved binder
takes effect upon evaporation of the solvent. In some cases a finely ground solid binder material
may be dispersed in the binder liquid thereby producing a cement-like solid binding bridge upon
evaporation of the solvent (Rhodes, 1998). In any of the three cases, the initial forming of the
liquid bridge is of primary importance regarding the properties and spatial structure of the final
agglomerate, and it is almost always the case that the solid bridge will have the form of the liquid
bridge (Summers & Aulton, 2001).
The particles are held together by liquid bridges at their contact points in the pendular state. This
situation requires that the saturation is low enough to let discrete binary bridges exist between the
solid surfaces. Such a lens-shaped ring of liquid cause adhesion due to the surface tension forces
of the liquid/air interface and the hydrostatic suction pressure in the liquid bridge (Summers &
Aulton, 2001). The capillary structure occurs when a granule is saturated. All the voids between
the particles are filled with binder liquid and the surface liquid of the agglomerates is drawn back
from the surface into the interior of the agglomerate. The particles are held together in this
configuration due to capillary suction at the liquid/air interface, which is now only at the
agglomerate surface. The funicular structure is a transition between the pendular and the capillary
state where the voids between the particles are not fully saturated. The droplet structure occurs
when the particles are held within or at the surface of a liquid binder droplet (Jain, 2002 and
Iveson et al., 2002). This situation almost never happens in fluid beds (Kunii & Levenspiel, 1991
and Litster & Ennis, 2004). A sketch of the different formal spatial agglomerate structures can be
seen in figure 1.
Figure 1: Spatial agglomeration structures.
The different formal spatial structures of liquid-bound agglomerates depending of liquid saturation
(Iveson et al., 2001a).
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
During granulation, it is possible for the saturation state of the agglomerates to change from e.g.
the pendular state to the droplet state either due to the continuous addition of liquid binder or the
saturation of the pores of each particle. The pendular structure is however the most common in
fluid bed granulation due to the usual processed particle sizes versus the droplet sizes (Kunii &
Levenspiel, 1991 and Litster & Ennis, 2004). Typical examples of agglomerates being bound by
solidified pendular liquid bridges can be seen in figure 2 in which Na2SO4 cores agglomerated
during coating with a Na2SO4/Dextrin solution. Hence, regarding the agglomeration process in
fluid beds, the primary attention should be given to the modelling of the pendular liquid bridge.
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Figure 2: Pendular bonding bridges.
Examples of agglomerates being bound by pendular bonding bridges (Hede, 2005).
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D
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Liquid-bound granule strength is dominated by three types of forces being interparticle forces,
static strength forces and dynamic strength forces (Iveson et al., 2001a). The first two are
interrelated as the tensile force of the liquid bridge acts to pull particles together and this normal
force at particle contacts activates friction. Static strength forces as surface tension and capillary
forces are conservative forces in the sense that they always act to pull particles together in wetted
systems. Frictional and viscous forces are dissipative as they always act against interparticle
motion. The complex interaction of these different forces means that it is often impossible in
industrial situations to predict a-priori the effect of changing any particular granule property,
unless the precise magnitude of each of these three types of forces is well-known (Iveson et al.,
2002). Hence the following sub sections will introduce the available theory regarding each of the
three types of forces, primarily in respect to the most important agglomerate structure being the
pendular liquid bridge.
1.3.1.1 Interparticle forces
The normal force generated by liquid bridges at inter-particle contacts activates inter-particle
friction forces. There are a whole range of different types of interparticle forces including van der
Waal´s forces, forces due to absorbed liquid layers, electrostatic forces and friction forces (Yates,
1983). According to Rumpf (1962) and Rhodes (1998) the friction forces and the forces
associated with the liquid/solid bridges (static and dynamic forces) are nevertheless the only
forces of significance in wet systems with particle sizes above roughly 10 m. This can be
visualised from figure 3 illustrating the relative magnitude of the different interparticle forces as
function of particle size.
Figure 3: Agglomeration strength overview.
Theoretical tensile strength of agglomerates with different bonding mechanisms (based on
Rumpf, 1962 and Rhodes, 1998).
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Internal friction forces inside agglomerates are often described in terms of a Columbic
relationship according to (Nedderman, 1992):
f
f n c
(1.5)
in which f is the macroscopic shear stress at failure, f the coefficient of internal friction, n the
macroscopic normal stress and c the cohesivity being in other words the shear strength at zero
normal load. This simple model represents the cumulative effect of several contributing
mechanisms. In an ensemble of particles in an agglomerate, resistance to shear deformation arises
from true tribological interaction between touching particles and particle interlocking of which
the last is a macroscopic locking mechanism depending on size and shape of the particles
constituting the agglomerate (Iveson et al., 2002). Generally, the amount of reported work
regarding internal friction forces in the presence of a viscous binder is very limited. Some
advances has been made in the understanding of the fundamental nature of the tribological
component of the friction force and these results indicate that the presence of a lubricating film
binding the particles together lowers the friction at particle as well as at the macroscopic level
(Cain et al., 2000). Iveson et al. (2002) present a short review of the latest advances in the field of
friction forces. Important trends to note from this review are that internal friction forces inside an
agglomerate are dissipative and not strongly strain-rate dependant at strain rates commonly
experienced in fluid bed granulation processes (Iveson et al., 2002).
1.3.1.2 Static strength – capillary and surface tension forces
The static strength of a pendular liquid bridge consists of two components. There is a suction
pressure caused by the curvature of the liquid interface and a force due the interfacial surface
tension acting around the perimeter of the bridge cross-section7. In the absence of gravitational
effects8 arising from bridge distortion and buoyancy, the total force F acting between two
spherical particles both of radius Rp is given by the sum of two components being the axial
component of the surface tension acting on the three-phase contact line and the hydrostatic
pressure acting on the axially projected area of the liquid contact on either particle. This leads to
the following expression commonly known as the boundary method (Lian et al., 1993):
Fpendular,boundary
˜ P ˜ R p ˜ sin 2 (M ) 2 ˜ lv ˜ R p ˜ sin(M ) ˜ sin(M )
2
(1.6)
in which is the half-filling angle, the contact angle and lv the liquid surface tension. The
situation may be seen formalised according to figure 4:
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Figure 4: Formal bonding schematic of an agglomerate.
Schematic representation of a liquid bridge of volume Vbridge between two spheres both of radius
Rp separated by a distance 2S with a neck radius of rneck, a liquid-solid contact angle T and a half
filling angle of (Based on Iveson et al., 2002 and Willet et al., 2000).
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P is the pressure difference across the curvature of the air-liquid interface9 and assuming that the
liquid surface has constant mean curvature, P is given by the Laplace-Young equation
(Goodwin, 2004 and Fairbrother & Simons, 1998):
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
P
ª1 1 º
lv « »
¬ r1 r2 ¼
(1.7)
in which r1 is and r2 are local principal radii of normal curvature in orthogonal directions at points
along the meridional bridge profile as shown in figure 4 (Iveson et al., 2001a). Conventionally, r2
is evaluated in a direction orthogonal to the meridional profile such that its center of curvature is
located along the surface normal on the bridge axis of symmetry. Hence, the center of curvature
of r2 is always within the bridge, or one of the spheres, and r2 itself is positive and equals rneck
when calculated at the neck point. r1 is evaluated in a direction containing the meriodional bridge
profile and can be positive or negative depending on whether or not its center of curvature lies
inside or outside the liquid bridge. A positive value of the parentheses in equation 1.7 leads to a
positive pressure difference P and thereby an attractive contribution to the pendular force in
equation 1.6 (Willet et al., 2000). The values of r1 and r2 can be evaluated at any point along the
bridge profile r(x) as it can be shown that the Laplace-Young equation can be rewritten according
to (Willet et al., 2000 and Iveson et al., 2001a):
P
§
¨
d 2r
¨
1
dx
lv ¨
3/2
½
¨§
2
2
·
ª
º
§
·
dr
§
·
dr
§
·
¨ ¨1 ¨ ¸ ¸
« r ˜ ¨1 ¨ ¸ ¸ »
¨ ¨ © dx ¹ ¸
«¬ ¨© © dx ¹ ¸¹»¼
¹
©©
·
¸
¸
¸
¸
¸
¸
¹
(1.8)
The determination of the exact surface profiles requires complex numerical procedures even for
the simple case of equally sized spheres and equation 2.8 cannot be solved analytically. This has
led to the use of approximate geometries such as circular (torodial) and hyperbolic arcs, which
have been shown to result in errors in bridge areas and volumes of the order of 1 % (Simons et al.,
1994). This has further led to a debate as to whether the surface tension and capillary pressure
should be evaluated at the mid-point of the liquid bridge where r2 = rneck instead of being
evaluated at the surface of the particles as it is the case with the boundary method. The prior
principle is often referred to as the gorge method and according to Lian et al., (1993), this
approach will give an expression for the pendular force according to:
Fpendular, gorge
˜ P ˜ rneck 2 ˜ ˜ rneck ˜ lv
2
(1.9)
in which rneck is the pendular bridge neck radius according to figure 5.
Whereas results by Hotta et al. (1974) supports the use of a slightly modified boundary method,
other results by Lian et al. (1993) indicates that the gorge method is the most precise and versatile.
Attempts of approximation of the total pendular force instead of relying on an analytical bridge
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
profile solution have been reported by Willet et al. (2000), and one example for a small pendular
bridge volume between two spheres is:
Fpendular, equally sized spheres
2 ˜ R p ˜ lv ˜ cos
§ S2 R p
1.0 2.1¨
¨ Vbridge
©
½
·
§ S2 R p
¸ 10.0¨
¸
¨ Vbridge
¹
©
·
¸
¸
¹
(1.10)
in which Rp is the radius of the equally sized spheres according to figure 5.
Figure 5: Liquid bridge bonding two primary particles.
A schematic representation of a small liquid bridge between two equally sized spheres where the
radius of curvature of the bridge surface is approximately given by the half-separation distance S
(Willet et al., 2000).
Interestingly, it can be seen from equation 1.10 that the pendular bridge force is directly
proportional to the liquid adhesion tension (Jlv ˜ cos T). This is in accordance with the wetting
thermodynamic studies reviewed by Hede (2005).
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Other expressions for equally sized spheres can be found in Israelachvili (1992). Rabinovich et al.
(2005) have suggested one of the newest expressions according to:
Fpendular, equally sized spheres
2 ˜ R p ˜ lv ˜ cos
1 (S/d sp/sp )
2 ˜ R p ˜ lv ˜ sinM ˜ sin( M )
(1.11)
in which:
d sp/sp
S ˜ 1 1 Vbridge /(2 ˜ R p ˜ S2 )
(1.12)
There is a general lack of reliable theoretical formulas for the calculation of the pendular force
between two unequal spheres although as an approximation, the radius term Rp in the previously
presented equations may be replaced by an effective particle radius Reff according to the
following equation (Rabinovich et al., 2005):
R eff
2R 1 R 2
R1 R 2
(1.13)
Granule static strength decreases as binder surface tension is lowered in respect to the surface
tension of the solid particle material. This is because the capillary suction pressure P as well as
the surface tension forces are both proportional to the liquid surface tension lv (Rumpf, 1962 and
Iveson, 2001). Likewise is it also expected that granule strength will decrease as the contact angle
increases due to the decreased wetting as emphasised in Hede (2005 & 2006b).
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16
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Static strength of liquid-bound granules is commonly measured and described in terms of tensile
static strength. Pioneer work by Rumpf (1962) in the sixties has lead to a generally accepted form
for the pendular bridge according to (Pierrat & Caram, 1997 and Ennis & Sunshine, 1993):
t, p
k cn
(1 ) Fpendular, boundary
2
dp
(1.14)
in which kcn is the coordination number being a function of the particle packing density10, is the
particle void fraction, dp the particle diameter and Fpendular, boundary is given by equation 1.6. The
model is based on the assumption that all of the powder particles are monosized spheres
distributed uniformly in the agglomerate on an average. Secondly, it is assumed that the pendular
bonds are, on an average, uniformly distributed over the surface and over the directions in space.
Finally, the effective pendular bonding forces are assumed to be distributed around a mean value,
which can be used in the calculations. These assumptions are generally accepted and several
empirical correlations based on the original Rumpf form exist. Schubert (1973 & 1975) reviews a
few suggestions being globally of the same form as equation 1.14. Pierrat & Caram (1997) have
fitted data from a range of different articles by different workers into a more general although
empirical relation:
7.80 ˜ (1 ) 3.03 ˜
t,p
1
˜ lv ˜ cos dp
(1.15)
In the case of capillary liquid bridges where all the pores are completely filled with liquid, the
interfacial forces exist only at the surface of the agglomerate and a negative capillary pressure
develops in the interior, holding the particles together. In that case, the static tensile strength is
given as (Schubert, 1975 and Pierrat & Caram, 1997):
t, c
Ssat ˜ a´˜
1 lv ˜ cos ˜
dp
(1.16)
in which a´ is a material constant taking a value of 6 for uniform spheres and a value between 6
and 8 otherwise, depending on sphericity, and Ssat is the saturation amount defined as the ratio of
the void volume occupied by the liquid to the total void volume (Pierrat & Caram, 1997).
The static tensile strength of a funicular liquid bridge can be related to t,c and t,p according to
(Pierrat & Caram, 1997) as:
t, f
t, p ˜
Sc Ssat
S Sf
t, c ˜ sat
Sc Sf
Sc Sf
(1.17)
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17
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
in which Sc and Sf are the upper saturation limits for the funicular/capillary transition and
pendular/funicular transition states respectively. Considerations by Flemmer (1991) indicate that
for randomly packed monosized particles, Sf is equal to 0.34 and Ssat is usually ranging between
0.25 and 0.50. Sc is usually above 0.90 (Pierrat & Caram, 1997).
The original Rumpf model as well as the models in equation 1.16 and in equation 1.17 predict
that granule tensile strength is proportional to the liquid surface tension and saturation, and that
t,c further increases with decreasing porosity and is inversely proportional to the particle
diameter. All of these trends have been validated by experimental data, but quantitatively the
models tend to over-predict the granule strength due to fact that these models fail to account for
the presence of extensive pore networks in the particles forming the agglomerate. As it has been
showed by Beekman (2000), failure often occurs by crack growth along such pore structures and
not by sudden failure across the whole liquid bridge plane. Some of the newer tensile strength
models try to account for this problem but as still most of the work on tensile strength of liquid
bound granules is performed at slow and invariant strain rates, the applicability of the above
presented equations is still not accurate enough for quantitative granulation purposes. This is due
to the fact that it is the amount of impact deformation and breakage that is critical in determining
agglomerate growth and breakage behaviour in fluid bed granulation equipment (Beekman, 2000
and Hede, 2005). That means that dynamic forces may become significant in the fluid bed
granulation process, especially when viscous binders are involved. Hence, the following sections
focus on the dynamic strength of liquid bridges.
1.3.1.3 Dynamic strength - viscous forces
The dynamic strength of a liquid bound granule can be approximated using lubrication theory as a
viscous force Fvis, which according to Lian et al. (1998) and Iveson et al. (2001a) may be
expressed as:
Fvis
6 ˜ liq ˜ rharm ˜ u 0 ˜ (1.18)
in which u0 is the relative initial velocity of the two particles, liq the liquid viscosity and is a
parameter being a function of the harmonic mean radius11 of the two particles rharm and the
separation distance between the two spheres H. For small separation distances between two
spheres in an infinite fluid, may be found as (Weinbaum & Caro, 1976):
1
2rharm
H
(1.19)
A more general expression has been derived by Adams & Perchard (1984) using classical
lubrication theory for flow between two spheres according to:
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18
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
rharm
2H
(1.20)
None of the two mentioned expressions for are appropriate to calculate precisely the viscous
forces as the underlying assumption that the particles move in an infinite fluid does not hold.
Instead, Jen and Tsao (1980) have found another expression for based on the half-filling angle according to:
½ ½ ˜ cosM
(1.21)
The expression for is thereby independent of the separation distance H which is in contrast to
experimental results reported by other authors (Ennis et al., 1990 and Iveson et al., 2001a).
Acknowledging that liquid bridges also have a viscous resistance to shear strain besides the
previously solely assumed axial strain, Goldman et al. (1987) derived another relationship based
on lubrication theory for small separation distances according to12:
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Fvis
·
§ 8 §r ·
6 ˜ liq ˜ rharm ˜ u 0 ¨¨ ln¨ harm ¸ 0.9588 ¸¸
¹
© 15 © H ¹
(1.22)
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19
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Results by Mazzone et al. (1987) indicated that dynamic liquid bridges are much stronger than
geometrically identical static liquid bridges in which the attraction forces is due to surface tension
only, as presented in the previous section. It was seen that the force required to separate two
moving particles is often significantly higher than that required in static situations because the
viscosity of the liquid resists the motion in the dynamic case. These observations help to explain
why even small amounts of liquid can drastically change the properties and fluidisation behaviour
during e.g. fluid bed coating.
1.3.1.4 Breakage and attrition of wet granules
The granule breakage phenomena may in principle be divided into the breakage of wet and
breakage of dry granules. Whereas dry granule breakage is often treated as a separate discipline
often without much attention to the process conditions, wet granule breakage is closely related to
the previously introduced theory. Although much of the basic theory and parameters regarding
the mechanical strength and breakage mechanisms are similar, there are a number of differences
between the breakage of a liquid bridge and the breakage of a solid amorphous or crystalline bond
or layer. The mechanical properties and breakage phenomena of dried granules and coating layers
were extensively covered in Hede (2005 & 2006b) and more introductionary information can be
found in Beekman (2000) as well as in a comprehensive recent review by Reynolds et al. (2005).
The topics on breakage and attrition of wet granules were however only slightly covered and
more on this topic will be presented below.
In the last step of the granulation process, the agglomerates grow too large to resist the agitative
motions in the bed. Coalescence will be counteracted by either an attrition of debris of the granule
surface either partly or totally, or by fracture of the agglomerate bonding. This is due to
insufficient binder distribution or simply due to the static stress in the drying liquid bridges
(Iveson et al., 2001a). A total fracture of wet or dry granules is seldom a problem in fluid bed
granulation however. The low shear properties in the bed rarely give sufficient kinetic energy to
fracture the granules completely (Waldie, 1991 and Schaafsma, 2000b). Simulations as well as
experiments performed by Khan & Tardos (1997) indicate that agglomerates are often broken
upon deformation by stretching when being sheared. They further showed that the stability of wet
agglomerates is closely related to the Stokes deformation number Stdef and that two regimes exist
involving high and low deformation characteristics based on the Stdef number. Khan & Tardos
(1997) defined the Stokes deformation number according to:
St def
m aggl ˜ u 02
(1.23)
2 ˜ Vaggl ˜ ( )
in which maggl is the mass of the agglomerate, Vaggl the volume of the agglomerate and
() is
some
characteristic stress in the agglomerate. In the most general case, this stress can be taken
according to the Herschel-Bulkley fluid model13 being:
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20
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
( )
y app ˜ n fi
(1.24)
where y is the yield strength, app is an apparent viscosity, nfi the flow index and the shear rate
(Tardos et al., 1997 and Fu et al., 2004). The Stokes deformation number defined in equation 1.23
increases with increasing particle size and reaches at some point during granulation a critical
value of St def * above which the agglomerates start to deform and eventually break. The critical
Stokes deformation number is not as well-defined as the critical viscous Stokes number
introduced in Hede (2005 & 2006b). This is due to the fact that, from a rheological point of view,
the system consisting of particles bound by viscous liquid bridges is a complex system that
exhibits both yield strength as well as non-Newtonian behaviour. Assuming that the agglomerate
is a very concentrated slurry of the binder material and the original particles, a first assumption is
that the apparent viscosity app is negligible compared with the yield strength meaning
that ( ) | y . It is further assumed that in fluid beds, the collision velocity u0 can be approximated
according to14:
u 0 | raggl ˜ (1.25)
These approximations all in all lead to a rough estimate of a theoretical expression for the critical
Stokes deformation number according to:
St
*
def
*
m aggl ˜ (rdef
˜ ) 2
2 ˜ Vaggl ˜ y
(1.26)
in which rdef* is the critical radius of the agglomerate after which deformation and breakage
occurs. Equation 1.26 thereby predicts an inverse linear relationship between the critical
agglomerate radius and the shear rate, which in fact has been observed in simulations as well as
experiments by Tardos et al. (1997) and Khan & Tardos (1997).
There is generally very limited experimental work on fracture and attrition of wet granules in
fluid beds as most work focuses on higher intensity mixer and hybrid granulators as drum and
high shear mixing (e.g. Fu et al., 2004). In fact the works of Tardos et al. (1997) and Khan &
Tardos (1997) are the only works so far specialised on fluid bed granulation. This is most likely
due to the fact that the high intensity granulation makes it much easier to estimate the average
shear forces and the collision velocity based on equipment parameters. Some of the newest
experimental studies by Fu et al. (2005) indicate conveniently that the failure patterns of wet
granules in high shear mixing equipment have a number of common features that have been
observed for dry granules including the formation of debris with conical geometry. Most of the
basic theory regarding mechanical strength and breakage mechanisms cannot however be applied
readily for wet granules.
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21
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Fu et al. (2005) observed that at small impact velocities, the agglomerates displayed homogenous
elastic, elastoplastic and plastic deformation. At some critical strain, stable meridian cracks
propagate from the impact zone in accordance with the theory emphasised in Hede (2005 &
2006b). At larger strains, chips were formed and the formation of larger fragments increased with
increasing impact velocity for the wet granules. It was seen additionally, that the critical impact
velocity for the formation of observable cracks increased with decreasing granule size and
increasing binder viscosity. Further it was concluded that agglomerates made with relative coarse
primary particles were generally more friable than agglomerates made with very fine primary
particles, and thus it was observed for the wet coarse particle agglomerates, that the critical
impact velocity decreased monotonically with increasing binder because of the reduction in the
fracture strength (Fu et al., 2005).
1.3.1.5 Wet granule strength summary
As emphasised in previous sections, the strength of agglomerates in the size range of roughly 10 –
1200 m is controlled by three main types of forces being static (capillary), frictional and viscous
(dynamic) forces. These forces are interrelated in a complex way and their relative importance
varies greatly with process and formulation conditions. This makes systematic experimental
investigations of wet granule strength a difficult subject although studies of wet granular strength
are somewhat more important than dry granular strength in respect to granulation (Fu et al., 2005).
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22
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
As it has been reviewed in earlier sections, traditional models do not include all three types of
forces, and e.g. the often quoted model by Rumpf (1962) consider only capillary forces. Such
simplified models may be sufficient for coarse particle systems held together by non-viscous
binders but must be considered invalid for agglomerates composed of fine particles bound by
viscous binders (Iveson et al., 2001a). That means that if any model should be quantitatively
applicable to industrial granulation processes, this model must include all three types of forces.
Developing models capable of predicting the complex interactions of capillary, viscous and
frictional forces that occur during wet granule impacts will be a challenging task. Some of the
latest results by Fu et al. (2005) indicate however that that impact failure mechanisms of wet
granules have a number of common features observed for breakage of dry granules including the
formation of debris with certain geometries etc. This is fortunate as the breakage of dry granule
traditionally has been more widely studied than the breakage of wet granules.
One of the modern approaches has been to try to simulate breakage of wet granules using
primarily discrete element method (DEM). The use of DEM in simulations seems to make some
progress in this field. E.g. have Lian et al. (1998) performed discrete element modelling
simulations of dynamic granule impacts. They simulated the collision of agglomerates in the
pendular states under conditions found in common fluid bed granulators. Results indicated that
granule strength is controlled by viscous and interparticle frictional energy dissipation with static
surface energies playing only a minor role. The use of DEM to simulate the breakage of wet
agglomerates is nevertheless a relative new scientific field and it still remains to be seen whether
this technique can be used to predict the complex effects of binder content, wetting behaviour and
other variables such as the influence of particle morphology on wet granule strength (Reynolds et
al., 2005 and Iveson et al., 2001a).
1.3.2 Granule growth modelling – Class I and Class II models
There are a large number of theoretical models available in the literature for predicting whether or
not the collision of two wetted particles will either result in permanent coalescence or just in
rebound. All models are associated with a number of assumptions and simplifications regarding
the mechanical properties of the particles as well as the system in which the particles collide.
Some of the first models were developed for predicting the sintering of fluidised beds used in the
mining industry, but later models have adapted to specifically describe the granulation process
(Iveson et al., 2001a). Although being very different in nature Iveson et al. (2001a) have divided
the granulation models into two classes of which the distinction principles can be sketched
according to figure 6.
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23
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Figure 6: Principles behind different classes of agglomeration models.
Schematic diagram of the two general classes of coalescence models: Class I models: Stick or
rebound, Class II models: Survival or separation (based on Iveson, 2001).
Class I models assume that the granules are free to move and that elastic properties of the particle
bodies are important. These models assume that initial coalescence occurs only if the kinetic
energy of collision is entirely dissipated and that otherwise the granules will rebound and move
apart. Various combinations of energy dissipation has been considered by different authors
including elastic losses, plastic deformation, viscous and capillary forces in the liquid binder and
adhesion energies of the contact surfaces. In class I models it is implicitly assumed that if the
initial impact results in permanent coalescence, then none of the subsequent impacts will be able
to break the two granules again. This means in other words that coalescence is assumed to occur
whenever the two particles do not possess sufficient kinetic energy to rebound (Iveson et al.,
2001a). It further implies that all collisions at near-zero velocities will result in permanent
coalescence (Iveson, 2001).
Class II models on the other hand assume that elastic effects are negligible during the initial
collision usually because it is assumed that the granules are plastic in nature and/or physically
constrained by surrounding granules. This leads to the simplification that all colliding granules
are in contact for a finite time t during which a liquid bridge develops between them. Permanent
coalescence thereby only occurs if this liquid bridge is strong enough to resist being broken apart
by subsequent collisions or shear forces. The strength of the binding bridge is assumed to be
dependent on factors such as the initial amount of plastic deformation and the length of time that
the two particles were in contact, which in other words means that it is assumed that the bonding
bridge will increase in strength as the liquid bridge turns into a solid bridge upon evaporation of
the binder solvent.
Due to the simplicity of both model classes, a lot of relevant critics are associated with both views
on modelling the agglomeration part of the granulation process. The coalescence criteria in Class
I models are typically criticised for being unreasonable for most fluid bed granulation
applications because this class of models neglects the effect of subsequent collisions. Two
granules which stick initially together may in reality be so weakly held together that they would
quickly break apart again. It is also unreasonable to assume that coalescence is controlled solely
by the initial collision energy, when in many applications the granules are constrained in contact
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24
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
with one another for significant lengths of time as it occurs in the quiescent zones of a fluid bed.
In these cases there is no single and uniform single collision event. Instead the granules are
constantly in contact with several others (Iveson, 2001). This means in other words that although
non-rebound is a necessary condition for permanent coalescence, it is not always a sufficient
condition. Not only do the particles need to stick when they first collide but they must also form a
bond strong enough to resist being broken by subsequent impacts in the fluid bed. This is not
accounted for in Class I models thereby being an obvious limitation.
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In all existing Class II models, only the first major separation event is considered and the
magnitude of separation is usually approximated by some global average value. If the particleparticle bonding bridge survives this single event then it is considered to be a permanent
agglomeration bond. However, in many granulators the separation events may have a wide range
of magnitudes and may further be distributed randomly in time. Therefore some criticisers state
that it is inappropriate to model coalescence by assuming a mean separation force which occurs at
regular intervals and that the probability of the survival of a bonding bridge rather will depend on
the history of impacts and the rate at which the binding bridges strengthens as they are kneaded
together by a number of low-level impacts (Iveson, 2001).
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25
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Class II models have generally been developed and optimised for high collision granulation
equipment such as high shear mixing or drum granulation. It is necessary to account for
deformability of the granules during collision as well as the rupture forces in such high agitative
equipment and a number of different class II coalescence models have been presented by e.g.
Ouchiyama & Tanaka (1975) for drum granulation. In fluid bed equipment, the agitative forces
and thereby the collisions are quite small which also implies that deformation of the particles
upon collision may often be neglected with good approximation. This means that assumptions
associated with Class I models have proven to be more accurate than those associated with Class
II models, and the latter type is almost never applied with fluid bed systems. Hence, only Class I
models will be presented here. For most industrial fluid bed purposes it is often adequate to model
the situation in which non-deformable particles collide and coalesce and only in rare cases is it
necessary to study coalescence of deformable particles, although it may be necessary in some
granulating fluid bed systems depending on the bed material. Both types of class I models will be
introduced in the following section.
1.3.2.1 Class I models: Coalescence of non-deformable granules
The gentle agitative forces in fluid beds mean that little permanent deformation usually occurs.
Under such low-agitative conditions granules coalesce by viscous dissipation in the surface liquid
before the granule core surfaces contact (Liu et al., 2000). As the two particles approach each
other, first contact is made by the outer liquid binder layer. The liquid will subsequently be
squeezed out from the space between the particles to the point where the two solid surfaces will
touch. A solid rebound will occur based on the elasticity of the surface characterized by a
coefficient of restitution15 e. The particles will start to move apart and liquid binder will be
sucked into the interparticle gap up to the point where a liquid bridge will form. This bridge will
either break due to further movement in the bed or solidify leading to permanent coalescence
(Tardos et al., 1997). In the described principle, granule coalescence will occur only if there is a
liquid layer present at the surface of the colliding particles. This growth principle continues until
insufficient binder liquid is available at the surface to bind new particles (Schaafsma et al., 1998).
The relative amount of binder liquid present at that stage is called the wetting saturation Sw and it
depends on the contact angle of binder liquid and the pore structure of the granule (Tardos et al.,
1997). The wetting saturation reflects the wettability of the particle and it is often approximated
by the binder droplet volume divided by the pore volume of a particle, under the assumption that
no drying occurs (Schaafsma et al., 2000a). Schaafsma et al. (2000a) have showed that Sw is
inversely proportional to the nucleation ratio J and the mean granule porosity g .
Ennis et al. (1991) have modelled the situation of coalescence in a fluid bed by considering the
impact of two solid non-deformable spheres each of which is surrounded by a thin viscous binder
layer. The simplified situation can be seen in figure 7.
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26
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Figure 7: Principles of the original Ennis et al. agglomeration model.
Schematic of two colliding granules each of which is covered by a viscous binder layer of
thickness h0 (Based on Ennis et al., 1991).
The principles of the model by Ennis et al. (1991) was introduced in Hede (2005 & 2006b) but
will be further emphasised in the present section. Being a typical Class I model it assumes
successful coalescence to occur if the kinetic energy of impact is entirely dissipated by viscous
dissipation in the binder liquid layer and only elastic losses in the solid phase. The model predicts
that collisions will result in coalescence when the viscous Stokes number (Stv) is less than a
critical viscous Stokes number (Stv*). The two numbers are given as (Ennis et al., 1991)16:
St v
8 ˜ g ˜ rharm ˜ u 0
9 ˜ liq
(1.27)
and
St *v
§ 1 · §¨ h 0 ·¸
¨1 ¸ ˜ ln¨
© e ¹ © h asp ¸¹
(1.28)
where Kliq is the liquid binder viscosity, e is the coefficient of restitution, Ug is the granule density,
h0 is the thickness of the liquid surface, hasp is the characteristic height of the surface asperities
and rharm is the harmonic mean granule radius of the two spheres given as (Iveson et al., 2001a):
rharm
2r1 r2
r1 r2
(1.29)
u0 is the initial collision velocity which is not easily obtainable due to the various phenomena
influencing the granule motion in fluid beds. A rough estimate based on the bubble rise velocity
Ubr has been presented by Ennis et al. (1991):
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27
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
u0 |
12U br rharm
db2
(1.30)
where db is the gas bubble diameter and G the dimensionless bubble space defined as the axial
fluid bed bubble spacing divided by the fluid bed gas bubble radius. Whereas the gas bubble
diameter and spacing can be estimated by the dimensions of the air distributor plate or found by
experiments, the bubble rise velocity is somewhat more difficult to determine. Davidson &
Harrison (1963) have however proposed the following empirical relation for a fluid bed based on
the bubble diameter db, gravity g, the minimum fluidisation velocity Umf and the superficial
velocity Us measured on empty vessel basis:
U s U mf 0.711 ˜ (g ˜ d b )1/2
(1.31)
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U br
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
The Stokes viscous number Stv can be seen as the ratio of kinetic energy to the viscous dissipation.
During fluid bed batch granulation Stv increases as the granules grow in size. This leads to three
possible situations. The first so-called “non-inertial regime” occurs when Stv << Stv*. All
collisions result in successful coalescence regardless of the size of the colliding granules, granule
kinetic energy or binder viscosity. As the granules grow larger the “inertial regime” occurs when
Stv | Stv*. The likelihood of coalescence now depends of the size of the colliding granules, and
granule kinetic energy and binder viscosity begin to play a role (Iveson et al., 2001a and Abbott,
2002). It can be seen from equation 1.29 and 1.27 that the collision between two small or one
small and one large granule is more likely to succeed in permanent coalescence than the collision
between two large granules due to the size of rharm and thereby the size of Stv versus Stv*. This is a
convenient way to understand why small particles agglomerate into larger ones (Tardos et al.,
1997). Eventually the system enters the “coating regime” when Stv >> Stv*. Here all collisions
between granules are unsuccessful and any further increase in the Stv will maintain the size of the
granules (Iveson et al., 2001a and Tardos et al., 1997). The existence of the three regimes has
been proved experimentally in different types of granulators (Ennis et al., 1991).
Granule growth is promoted by a low value of Stv and a high value of Stv*. For instance,
increasing the binder content will increase the binder layer thickness h0 which will increase Stv*
and hence increase the likelihood of successful coalescence. The effect of the binder viscosity is
not easily predictable in that e.g. increasing the value Kliq (lowering Stv) alters the coefficient of
restitution e decreasing Stv* as well (Iveson et al., 2001a).
Although the Stv and the Stv* are important parameters in the prediction of coalescence they are
only valid for predicting the maximum size of granules which can coalesce. The parameters state
nothing about the rate of granule growth. Different authors have showed however, that fast
growth rates are attributed to the non-inertial regime while a slower growth is attributed to values
of Stv close to or above Stv* (Ennis et al. 1991 and Cryer, 1999).
The model by Ennis et al. (1991) was a significant progress in the modelling of particle-level
coalescence in fluid beds as it was the first model to consider dynamic affects such as viscous
dissipation. Results by e.g. Ennis et al. (1991) and Hede (2005) indeed shows that agglomeration
tendency is closely related to the relative sizes of Stv* versus Stv . The model is nevertheless
limited by its many assumptions17 although is gives a rough number for the indication of the limit
between no-agglomeration and successful agglomeration. The model is however only valid for
non-deformable surface wet granules where the viscous forces are much larger than capillary
forces. These approximations will not always hold for all granulating fluid bed systems and
sometimes more advanced models must be used. The next section introduces an extension of the
original Ennis et al. (1991) model.
1.3.2.2 Class I models: Coalescence of deformable granules
Despite the gentle nature of fluid bed agitation compared to other granulating systems, some
particle materials will deform upon collision in fluid beds. Based on the original model by Ennis
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
et al. (1991), Liu et al. (2000) extended the viscous Stokes theory to include the effect of plastic
deformation of the granules upon collision. Granules are assumed to have a strain-rate
independent of Young’s modulus E and plastic yield stress Yd. Liu et al (2000) consider two cases
being surface wet granules and surface dry granules where liquid is squeezed to the granule
surfaces by the impact. In respect to fluid bed granulation and coating, only the first case is
relevant and will be presented in the following. Behind the model lies the assumption that
coalescence occurs when the kinetic energy of impact is all dissipated through viscous dissipation
in the liquid layer as well as through plastic deformation of the granule bulk. It is further assumed
that granule surfaces only deform when they come into physical contact, which may hold for fluid
bed situations but not in e.g. mixer equipment where the pressure generated by the coating fluid
being squeezed between the surfaces will cause some precontact deformation. Of further
assumptions should be mentioned that attractive interparticle forces in the contact area are
assumed to be negligible and that fluid cavitation does not occur during rebound (Liu et al., 2000).
The model extension by Liu et al. (2000) divides the coalescence phenomenon into two types –
type I and type II, which must not to be confounded with the coalescence model of Class I and
Class II. Type I coalescence occurs when granules coalesce by viscous dissipation by the surface
liquid before the granules are able to touch (Iveson et al., 2001a). That is, the granules are halted
and coalesce before their surfaces come into contact. Type II coalescence occurs when granules
are slowed to a halt during rebound, after their surfaces have made contact (Liu et al., 2000). In
type II coalescence the relative granule velocity is reduced to zero by viscous forces during
rebound after their surfaces have made contact. It is important to note that both deformable and
non-deformable granules can grow by either mechanism although deformable granules can
coalesce by type II over a greater range of Stv values (Liu et al., 2000).
The general simplified coalescence situation for both coalescence types can be illustrated by
dividing the particle collision situation into four stages. In figure 8 the collision between two
surface wet deformable granules is considered. The first stage is the approach stage. At a
separation distance of 2h0 the liquid layers touch and merge. This combined liquid layer will then
be squeezed by out as the granules approach, dissipating some of the kinetic energy of collision.
In the second deformation stage the granules will begin to deform as the separation distance
reduces to 2hasp, which is the height of the surface asperities, and the relative granule velocity is
reduced to 2u1. The remaining kinetic energy is converted to stored elastic energy and dissipated
by plastic deformation. When the relative collision velocity is reduced to zero a contact area of A*
is formed between the granules. In the third separation stage the granules begin to rebound with
an initial velocity of u2 as the stored elastic energy is released. Viscous dissipation in the surface
liquid layer will again retard the granule movement. In the fourth last separation stage the liquid
layers are assumed to separate and the granule rebound to be complete when the granules are
separated to a distance of 2h0 leaving the granules with a velocity of u3.
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30
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Figure 8: Stages in the agglomeration process of deformable primary particles.
Schematic diagram of the model used to predict coalescence of two surface wet deformable
granules. A) Approach stage. B) Deformation stage. C) Initial separation stage. D) Final
separation stage (Based on Iveson et al., 2001a and Liu et al., 2000).
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Following the terminology of figure 8, type I coalescence occurs if the viscous forces are so high
that the granule velocity u1 is less than zero indicating that coalescence occur before the granule
surfaces come into contact. In such a situation, permanent coalescence will occur if the following
condition is satisfied:
§ h ·
St v ln¨ 0 ¸
¨h ¸
© asp ¹
(1.32)
in which the viscous Stokes number is defined according to equation 1.27.
For granule collision situations with u1 > 0 at the separation distance of 2hasp, the granule surfaces
come into contact and begin to deform. For granules to permanent coalescence u3 must obviously
be less than zero. The following condition must be satisfied for permanent coalescence to occur
(Liu et al., 2000):
½
§
··
§
¨1 1 ln¨ h 0 ¸ ¸
¨ St v ¨ h asp ¸ ¸
¹¹
©
©
·
§ h 0 ··
2h 02
¸¸ ˜
¸
¨
1 ln
¸ ( ) 2 ¨ h ¸ ¸
pdef
¹
© asp ¹ ¹
0.172 § 2rharm ·
§ Yd ·
¨
¸
¨ * ¸ ˜ (St def ) 9/8 St v ¨© h 0 ¸¹
©E ¹
§§ h 2
·
¨ ¨ 0 1¸ 2h 0
2
¸ pdef
¨ ¨ h asp
¹
©©
§ h0
¨
¨h
© asp
2
§
· ¨§
§
1 §¨ h 0 ·¸ ·¸
§Y ·
¨¨1 7.36 ˜ ¨ d* ¸(St def ) 1/4 ¸¸ ˜ ¨ ¨1 ln
©E ¹
¹ ¨ ¨© St v ¨© h asp ¸¹ ¸¹
©
©
½
·
¸
¸¸
¹
5/4
˜
(1.33)
2
in which pdef is the extent of permanent plastic deformation given by:
pdef
½
§
1 §¨ h 0 ·¸ ·¸
§ 8 ·
½
˜
ln
¨ ¸ ˜ (St def ) ˜ 2rharm ¨1 ¨ St v ¨ h asp ¸ ¸
© 3 ¹
©
¹
©
¹
½
§
§
1 §¨ h 0 ·¸ ·¸
§ Yd ·
1/4 · ¨
ln
˜ ¨¨1 7.36 ˜ ¨ * ¸(St def ) ¸¸ ˜ ¨1 ¸
©E ¹
¹ © St v ¨© h asp ¸¹ ¹
©
(1.34)
and rharm is the harmonic mean radius given by equation 1.29, Yd is the plastic yield stress and E*
is the granule Young’s modulus given by:
1
E*
1 12 1 22
E1
E2
(1.35)
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
in which E1 and E2 are the Young’s modulus of the two granules and 1 and 2 the Poisson ratios18
respectively. Analogously19 to the definition of the Stdef in equation 1.26, the Stokes deformation
number is in this case expressed as (Tardos et al., 1997):
St def
m harm ˜ u 02
2 ˜ (2 ˜ rharm )3 ˜ Yd
(1.36)
where mharm is the harmonic mean mass according to:
m harm
2m 1 ˜ m 2
m1 m 2
(1.37)
The Stokes deformation number can be seen as the ratio of impact kinetic energy to plastic
deformation in the granule core. It gives a measure of the amount of plastic deformation that the
granules would have suffered during the collision had there been no viscous liquid layer present.
Based on the criteria for type I and type II coalescence, a plot of Stv versus Stdef indicating the
boundaries between the two types of coalescence has been suggested by Liu et al. (2000)
according to figure 9:
Figure 9: Class I coalescence.
Values of the Stokes viscous number Stv versus the Stokes deformation number Stdef showing
regions of rebound and coalescence for wet deformable granules (Liu et al., 2000).
Figure 9 suggests that three primary regions exists being rebound, type I and type II regions. In
accordance with the models in equation 1.27 and equation 1.36, figure 9 suggests that at low Stdef
number the likelihood of coalescence depends only on the critical viscous Stokes number Stv* as
it was suggested by the original model for non-deformable granules by Ennis et al. (1991). In this
region all collisions are fully elastic. As Stdef increases, the coalescence region extends over a
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33
Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
wider range of Stv values. This is because permanent granule deformation occurs which improves
the likeliness of permanent coalescence in two ways: First of all it dissipates some of the impact
energy and secondly, it creates a flat surface of area A* between the two granules which creates a
greater viscous dissipation force during rebound (Iveson et al., 2001a). In comparison to the
original model by Ennis et al. (1991), the extension by Liu et al. (2000) results in a model that
predicts that when plastic deformation is significant, increasing the impact velocity may actually
improve the likelihood of coalescence by shifting a system from rebound back into the
coalescence region. This situation is rarely observed in fluid beds however, as the low agitative
forces result in coalescence domains in which Stdef and Stv numbers are both small.
Despite the improvement of the original Ennis et al (1991) model, the model by Liu et al. (2000)
still suffers many of the same limitations. E.g. are capillary forces still neglected although these
are acknowledged having an importance. Likewise has no account been taken of any possible
breakage of chipping during collision. The Liu et al. (2000) model is however most likely
currently the best available model to describe coalescence situations at particle level, and
validation experiments with glass ballotini cores in drum granulation equipment seems promising
(Liu et al., 2000).
1.3.3 Granule growth regimes
With basis in the Stokes deformation number in equation 1.36 Iveson & Litster (1998) proposed
that there are basically two broad categories of granule growth behaviour being steady growth
and induction growth. At steady growth the size of the agglomerates increases linearly with time
while at the induction growth there is a delay period during which little growth occurs.
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Steady growth occurs in systems with weak and deformable granules. Granules in this regime
grow by either crushing or deformation, thereby creating a large area during impact which
promotes the chance of permanent coalescence. Steady growth is generally exhibited by relatively
coarse, narrowly sized particles. Induction growth occurs in systems which are relatively strong.
The granules do not deform sufficiently during impact to be able to coalesce without the presence
of liquid binder at the particle surfaces. Hence, after the initial nuclei form, there is a delay
period during which little growth. If the granules consolidate sufficiently to squeeze binder liquid
to the surface, then the granules will begin to grow quickly until a critical size is reached above
which the torque experienced by the dumbbell particle pairs becomes too large for further
coalescence growth. Induction growth is generally seen in systems with fine widely spread
particles and/or viscous binders. Whereas steady growth has been observed in fluid bed
granulating systems, induction growth has not yet been reported in fluid beds probably because
the impact forces are too low to cause significant consolidation as emphasised in Hede (2005 &
2006b).
Iveson & Litster (1998) suggested that the type of granule growth behaviour is a function of two
parameters only being the amount of granule deformation during impact according to equation
1.36 and the maximum pore liquid saturation being a measure of the liquid content according to:
s max
w mr ˜ p ˜ (1 min )
b ˜ min
(1.38)
in which wmr is the mass ratio of liquid to solid, p is the density of the solid particles, b is the
density of the binder liquid and min is the minimum porosity the formulation reaches for that
particular set of operating conditions. The liquid saturation terms smax must include any extra
liquid volume due to solids dissolution but should not include any liquid which is absorbed into
the porous carrier particles (Iveson et al., 2001a and 2001b).
Based on values of smax and Stdef, Iveson & Litster (1998) suggested a granule growth regime map
according to figure 10:
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Figure 10: Granule growth regime map (Iveson & Litster, 1998).
In the original regime map by Iveson & Litster (1998) a number of subcategories of steady and
induction growth were suggested. Nucleation growth only occurs when granule nuclei form, but
on the same time there is insufficient binder to promote further growth. Crumb behaviour occurs
when the binder conditions are too weak to form permanent coalesced granules but instead form a
loose crumb material which cushions a few larger granules which are constantly breaking and
reforming (Iveson et al., 2001a). Overwetting occurs when excess binder has been added and the
system forms an oversaturated slush or slurry (Iveson & Litster, 1998). The original regime map
does not include any values of the Stokes deformation number for which the regimes change.
Work by Iveson et al. (2001b) suggested a modified granule growth regime map in which the
location of several of the regime boundaries has been identified quantitatively. Iveson et al.
(2001b) suggested that the crumb and slurry regions occur at Stdef > 0.1 with slurries when smax >
100%. The transition from nucleation to induction growth also occurs at smax = 100%. The smax at
which steady growth begins is shown to decrease with increasing Stdef. In accordance with these
observations, Iveson et al. (2001b) proposed a modified regime map according to figure 11.
Figure 11: Modified granule growth regime map (Iveson et al., 2001b).
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Fluid bed agglomeration at particle level
Advanced granulation theory at particle level
Although being a step towards a quantitative prediction of the agglomeration phenomena, the
granule growth regime map is still more of a descriptive tool and not a predictive one. This is due
to the fact that the two parameters smax and Stdef require a-priori knowledge of the maximum
extent of consolidation since this affects granule yield stress as well as pore saturation. Another
significant shortcoming is the very simplistic rheological model used to describe the mechanical
properties of the granules. It is assumed that the granules are rigid-plastic materials whereas in
reality as previously stated, wet granular materials are complex visco-elastic-plastic materials
with strain-rate and history dependent behaviour (Iveson et al., 2001a). Subsequent work by
Iveson et al. (2001b) suggests that binder viscosity20 needs to be included as a third independent
parameter in the granule growth regime map thereby expanding the map into three dimensions.
The regime map boundaries also suffer from the fact that it is associated with great difficulty of
comparing different types of equipment using the regime map because of the uncertainty of the
correct characteristic granule impact velocity (Iveson et al., 2001a). A suggestion for further work
could be to develop an equipment specific regime map as not all regimes in the current maps are
observed for all types of equipment.
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Summary
Advanced granulation theory at particle level
Summary
The modelling of the different processes that occur during granulation at particle level may at first
seem a too simplified approach in the process of being able to describe the entire granulating
system. Much of the theory presented in the present text takes a starting point in the situation
where only two particles interact and in addition only considers a situation in which only one of
many types of processes occurs. Further, a lot of assumptions are associated with the theory, and
a-priori knowledge is often needed in order to apply the equations quantitatively. The situation
inside e.g. a fluid bed during granulation is obviously far more complex and many of the
parameters needed in the micro-level equations cannot be determined easily or if they can, the
single needed property may in fact consist of a wide distribution of values rather than a single
value. That is e.g. the case with the particle collision velocities u0 as previously emphasised.
Hence, much of the theory presented may most likely not be applied readily for real granulating
systems for other than qualitative purposes.
Micro-level modelling is nevertheless a vital tool towards a complete engineering of the
granulation process at all scales. Although the introduced approaches are somewhat too simplified,
they form a vital part in the process towards a fundamental understanding of the granulation
process. Only by simplifications is it possible to study the various granulation processes and
phenomena in detail. As it is the present case that meso-scale properties are still unknown,
detailed micro-level theory is a prerequisite in the modelling of the granulation process at larger
macro-level scale, and as it has been presented elsewhere (e.g. in Hede, 2006a), micro-level
understanding play an important role in the different simulation techniques as well as population
balance modelling of the fluid bed granulation process.
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Table of symbols
Advanced granulation theory at particle level
Table of symbols
Nomenclature
Unit (SI-system)
a
a´
ad
aAE
an
A
Internal coordinate
Material constant
Projected area of liquid binder droplets
Fitting parameter
Projected area of a nucleus granule
Powder flux
Dimensionless
m2
Dimensionless
m2
m2/s
A*
b
bAE
c
dair distrib pl.
db
dbed
dd
dd,rel
dp
dorifices
dsp/sp
dv
dvessel
e
E
E*
Eelu
f1(x, r, t)
fbi
finitial
ftetra
Fpend.,bound.
Fpend.,eq sph.
Fpend.,gorge.
Fvis
Fi
Fi H
Fi E
Fi P
g
Contact area between colliding granules
Internal coordinate
Fitting parameter
Cohesivity of dry particle mass
Air distribution plate diameter
Gas bubble diameter
Fluidised bed diameter
Liquid droplet diameter
Relative liquid droplet diameter
Particle diameter
Pitch orifice diameter
Interaction parameter of two spheres
Equivalent diameter of particles
Fluid bed vessel diameter
Particle coefficient of restitution
Young modulus
Granule Young modulus
Elutriation rate
Average number density function
Bi-variant average number density function
Initial average number density function
Tetra-variant average number density function
Pendular force in the “boundary” method
Pendular force between two equally sized spheres
Pendular force in the “gorge” method
Viscous force
Net force vector acting on particle i
Drag force vector
Force vector accounting for external fields
Force vector accounting for particle-particle interactions
Gravity
m2
Dimensionless
N/m2
m
m
m
m
m
m
m
m
m
m
Dimensionless
N/m2
N/m2
N
N
N
N
m/s2
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Table of symbols
Advanced granulation theory at particle level
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G(x, r, t)
Gs
h(x, r, t)
h0
ha+
hahasp
hb+
hbhbed
H
i
Ii
J
k
k´
kcn
Ka
L
Lbed
Lslr
Rate of growth by layering
Mass flux of particles
Net generation rate of particles
Binder layer thickness covering colliding granules
Birth of particles due to aggregation
Death of particles due to aggregation
Characteristic length scales of surface asperities
Birth of particles due to breakage
Death of particles due to breakage
Bed height
Separation distance between two spheres
Summation parameter
Moment of inertia
Nucleation ratio
Proportionality constant
Proportionality constant
Coordination number
Nucleation area ratio
Characteristic length of particles
Fluid bed length from distributor plate to exhaust exit
Length scale ratio
m2/s
m
m
m
m
Dimensionless
Dimensionless
Dimensionless
m
m
Dimensionless
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40
Table of symbols
Advanced granulation theory at particle level
m
m(x)
maggl
mbed
mharm
nozzle air
m
Mass
Mass of a particle of internal state x
Agglomerate mass
Bed load
Harmonic mean granule mass
Spray rate from the nozzle
kg
kg
kg
kg/s
spray
m
Air flow rate through the nozzle
kg/s
Mi
nfi
n(x, r, t)
Net torque vector
Flow index
Actual number density
Nucleation rate
Dimensionless
No. of particles/s
rg
Average total number of particles
Total number of particles
Initial total number of particles
Summation number
Pressure
Probability density function
Discretisation number
Radius
Radius of an agglomerate
Critical radius of an agglomerate after which def. occurs
Harmonic mean granule radius
Pendular bridge neck radius
Mean granule size
Dimensionless
Pa
Dimensionless
m
m
m
m
m
m
rg 0
Initial mean granule size
m
Pendular bridge neck radius
External coordinate vector
External coordinate vector
Radius
Particle radius
Maximum pore liquid saturation
Distance
Saturation at transition funicular/capillary state
Dry coating material feed rate
Saturation at transition pendular/funicular state
Amount of saturation
Stokes deformation number
Critical Stokes deformation number
Viscous Stokes number
m
m
m
Dimensionless
m
Dimensionless
Dimensionless
Dimensionless
Dimensionless
Dimensionless
Dimensionless
n 0
N(r, t)
NT
NT0
p
P
P(x,r__x´,r´)
q
r
raggl
rdef*
rharm
rneck
rneck
r
r´
R
Rp
smax
S
Sc
Sd
Sf
Ssat
Stdef
Stdef*
Stv
kg
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Table of symbols
Advanced granulation theory at particle level
Stv*
Sw
S(q)
SKolmogorov
t
tcoat
u
u0
U
Ubr
Umf
Us
vi
v
V
Critical viscous Stokes number
Wetting saturation
Summation function
Kolmogorov entropy
Time
Coating time
Granule velocity
Initial granule collision velocity
Fluidisation velocity
Bubble rise velocity for a fluid bed
Minimum fluidisation velocity
Superficial gas velocity
Velocity vector
Particle volume internal coordinate
Average particle volume
Liquid binder volume internal coordinate
Volumetric spray rate
Dimensionless
Dimensionless
bits/s
s
s
m/s
m/s
m/s
m/s
m/s
m/s
m3
m3/s
Vaggl
Vbridge
Vr
Vx
w
w*
wmr
W
x
x´
x
y
Y(r,t)
Yd
z
Agglomerate volume
Liquid bridge volume
Volume of external coordinates
Volume of internal coordinates
Granule volume parameter in coal. kernel expression
Critical average granule volume
Mass ratio of liquid to solid
Spray zone width
Internal coordinate vector
Internal coordinate vector
Coordinate
Coordinate along the width of the spray zone
Continuous phase vector
Plastic yield stress
Counting number
m3
m3
Dimensionless
m
m
m
N/m2
Dimensionless
Coalescence kernel
Rate constant
Aggregation probability in time interval dt
Coefficient of interphase drag
Coalescence kernel expression
Dimensionless bubble spacing
Extent of permanent plastic deformation
Dimensionless
Dimensionless
Dimensionless
v
vL
Greek
0
dt
id
*
G
Gpdef
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Table of symbols
Advanced granulation theory at particle level
f
mean
f
n
width
t,f
t,p
t,c
( )
Vy
Wc
Wd
\a
\n(y)
\n
Dimensionless
m
Pa
Pa
m
N/m2
N/m2
N/m2
N/m2
Yield stress/strength
Average particle circulation time
Droplet penetration time
Dimensionless spray flux
Dimensionless nuclei distribution function
Dimensionless spray number
Particle shape factor (sphericity)
Dimensionless parameter in the dynamic strength eq.
Half filling radius
Contact angle
Poisson ratio
N/m2
s
s
Dimensionless
Dimensionless
Dimensionless
Dimensionless
Dimensionless
q
q
Dimensionless
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Coefficient of internal friction
Mean in the Gaussian distribution
Macroscopic shear stress at failure
Macroscopic normal stress
Standard deviation
Funicular bridge static tensile strength
Pendular bridge static tensile strength
Capillary bridge static tensile strength
Characteristic stress in an agglomerate
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Table of symbols
Advanced granulation theory at particle level
Q(x´,r´,Y,t)
H
Hlongitudinal
Hmin
Htrans
Average number of particles formed from break up
Particle voidage (void fraction)
Longitudinal extension strain
Minimum porosity
Transverse contraction strain
Mean granule porosity (void fraction)
%
Dimensionless
%
Dimensionless
%
Interfacial surface tension between liquid and vapour
Shear rate
N/m
s-1
U
Ub
Ug
Up
app
Kliq
i
H
r
x
Density
Binder liquid density
Granule density
Particle density
Apparent viscosity
Liquid (binder/coating) viscosity
Angular velocity vector
Hounslow discretisation parameter
Domain of external coordinates
Domain of internal coordinates
kg/m3
kg/m3
kg/m3
kg/m3
kg s /m
kg s /m
Dimensionless
Dimensionless
g
Jlv
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Literature
Advanced granulation theory at particle level
Literature
Abbott, A.: Boundary Between Coating and Granulation, Master Thesis, Department of Chemical
Engineering, The University of Queensland, 2002.
Adams, M.J. and Perchard, V.: The Cohesive Forces between Particles with Interstitial Liquid,
International Chemical Engineering Symposium Series, No. 91, pp. 147-160, 1984.
Adetayo, A.A., Litster, J.D., Pratsinis, S.E. and Ennis, B.J.: Population balance modelling of
drum granulation of materials with wide size distribution, Powder Technology, No. 82, pp. 37-49,
1995.
Adetayo, A.A. and Ennis, B.J.: Unifying Approach to Modelling Granule Coalescence
Mechanisms, AIChE Journal, No. 43, pp. 927-934, 1997.
Batterham, R.J., Hall, J.S. and Barton, G.: Pelletizing Kinetics and Simulation of Full-Scale
Balling Circuits, Proceedings 3rd International Symposium on Agglomeration, Nürnberg,
Germany, 1981.
Beekman, W.J.: Measurement of the Mechanical Strength of Granules, Ph.D. Thesis,
Technische Universiteit Delft, 2000.
Bell, T.A.: Challenges in the scale-up of particulate processes – an industrial perspective,
Powder Technology, No. 150, pp. 60-71, 2005.
Biggs, C.A., Sanders, C., Scott, A.C., Willemse, A.W. Hoffman, A.C., Instone, T., Salman, A.D.
and Hounslow, M.J.: Coupling granule properties and granulation rates in high-shear
granulation, Powder Technology, No. 130, pp. 162-168, 2003.
Boerefijn, R. and Hounslow, M.J.: Studies of fluid bed granulation in an industrial R&D context,
Chemical Engineering Science, No. 60, pp. 3879-3890, 2005.
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60
Endnotes
Advanced granulation theory at particle level
Endnotes
1
Please refer to Hede (2005) section 4.2.3.
2
More on Monte Carlo techniques may be found in Hede (2006a).
3
Please refer to Hede (2005) section 4.2.4.
4
The time it takes for a particle to circulate a complete wetting-drying cycle. It depends
primarily on the fluidisation velocity and bed height, as it was presented in Hede (2005 &
2006b).
5
Please refer to equation 4.11 in Hede (2005).
6
It can be shown mathematically that this does not affect the density of binder liquid along
the width of the spray zone (Wildeboer et al., 2005).
7
There may obviously also be buoyancy force due to the partial submersion of the particle
spheres but Princen (1968) showed that this is negligible for particle spheres with
diameters less than 1 mm.
8
Only large liquid bridges are affected by gravitational distortion (Simons et al., 1994).
9
Sometimes also referred to as the capillary suction pressure (Iveson et al., 2001a).
10
Ennis & Sunshine (1993) suggest a value for kcn of 1.1.
11
Please refer to equation 1.29.
12
Equation 1.22 has not yet been fully verified (Iveson et al., 2002).
13
The Herschel-Bulkley model is often preferred to power law or Bingham relationships as
it results in more accurate models on Non-Newtonian behaviour when adequate
experimental data is available (London, 2005).
14
Please note that this assumption is basically the same assumption presented by Ennis et al.
(1991) in which the collision velocity is based on the bubble rise velocity Ubr, bubble
diameter db, radius of the agglomerate raggl and the axial fluid bed bubble spacing .
Please refer to Hede (2005), pp. 22 for further information.
15
The coefficient of restitution is the ratio of the difference in velocity before and after the
collision. In the case of two colliding particles it is the difference in the velocities of the
two colliding particles after the collision divided by the difference in their velocity prior
to collision. Perfect elastic collisions has e = 1 (Christensen et al., 2000). The coefficient
of restitution e thereby accounts for the viscous dissipation in the binder phase being
sufficient to dissipate the energy of collision (Liu et al., 2000).
16
Please refer to appendix A4 in Hede (2006b) to see the derivation of equation 1.27 and
equation 1.28.
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17
For instance that there exists a uniform granule collision velocity or that in coalescence,
capillary forces can be neglected (Ennis et al., 1991). Ennis et al. (1991) justified
neglecting capillary forces on the basis that the energy added during liquid bridge
formation and granule approach is cancelled by the energy dissipated during granule
separation and liquid bridge rupture. This is a rough assumption as that dynamic energy
of the pendular bridge and the rupture energy are equal only if the collision has a
coefficient of restitution equal to one, which is almost never the case (Liu et al., 2000).
Especially the difficulty of determining a precise collision velocity u0 makes the viscous
Stokes theory difficult to exploit fully in practice (Abbott, 2002).
18
Poissons ratio is the ratio of transverse contraction strain to longitudinal extension strain
in the direction of the stretching force. Tensile deformation is considered positive and the
compressive deformation is considered negative. Hence, the Poisson ration is defined as:
= -trans / longitudinal (Wisconsin, 2005).
19
Please note that the superstructure of equation 1.36 in fact is the same as equation 1.23 in
which the harmonic mean mass has been substituted with the mass of the agglomerate,
the harmonic mean radius with the volume of the agglomerate and the plastic yield stress
with the characteristic stress in the agglomerate.
20
Characterised by the viscous Stokes number Stv (Iveson et al., 2001b).
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Endnotes
Advanced granulation theory at particle level